Simplify the following expression and state the condition under which the simplification is valid. You can assume that $r \neq 0$. $p = \dfrac{4(r - 5)}{6} \div \dfrac{5r(r - 5)}{r} $
Answer: Dividing by an expression is the same as multiplying by its inverse. $p = \dfrac{4(r - 5)}{6} \times \dfrac{r}{5r(r - 5)} $ When multiplying fractions, we multiply the numerators and the denominators. $p = \dfrac{ 4(r - 5) \times r } { 6 \times 5r(r - 5) } $ $ p = \dfrac{4r(r - 5)}{30r(r - 5)} $ We can cancel the $r - 5$ so long as $r - 5 \neq 0$ Therefore $r \neq 5$ $p = \dfrac{4r \cancel{(r - 5})}{30r \cancel{(r - 5)}} = \dfrac{4r}{30r} = \dfrac{2}{15} $